PRACTICAL PROBLEMS.

1. What is the g. c. d. of $18 and $45?

2. What is the l. c. m. of 12 ft., and 90 ft.?

3. A farmer has 225 bu. of oats, 135 bu. of wheat, and 90 bu. of rye, which he wishes to put in bins of equal size; each bin to be as large as possible; how many bushels must each hold that all may be filled without mixing the different kinds of grain ?

4. What is the shortest piece of wire that can be cut up into exact lengths of either 6 ft., 8 ft., or 10 ft. ?

5. There are three companies of soldiers containing respectively 36, 60, and 84 men, each of which is to be divided into platoons; how many men must be put in a platoon, so that all the platoons shall be equal and each contain the greatest possible number of men?

6. How many quarts are there in the smallest cask of cider that can be exactly measured by either a 3 quart, a 5 quart, or a 6 quart measure ?

(52.) What is a common divisor of two or more numbers? What is the greatest common divisor of two or more numbers? When are numbers prime with respect to each other? (53.) What general principles are used in finding the greatest common divisor? (54.) Give the rule for finding the greatest common divisor by the method of factors. (55.) What additional principle is used? (56.) Give the rule for finding the greatest common divisor by the method of continued division. How do you find the greatest common divisor of more than two numbers? (57.) What is a multiple of a number? What is a common multiple of two or more numbers? What is their least common multiple? (58.) Give the principles used in finding the least common multiple. (59.) Give the rule for finding the least common multiple.

60. If a unit is divided into equal parts, each part is called a fractional unit.

If the unit is divided into two equal parts, each is called one-half; if into three, each is called onethird; if into four, each is called one-fourth; and so on. Fractional units may be written and read as shown below:

one-half, one-third, one-fourth, one-fifth, one-sixth. one-seventh, etc.

The Reciprocal of a Number is 1 divided by that number. Thus, is the reciprocal of 2, is the reciprocal of 3, and so on.

MENTAL EXERCISES.

1. If a unit is divided into 5 equal parts, what is each part called? If it is divided into 9 equal parts, what is each part called? If into 10? If into 12? If into 13? 2. How many halves of an apple are there in 1 apple? How many fifths? How many ninths? How many tenths? How many twelfths? Fifteenths?

Twentieths?

3. What is the reciprocal of 10? Of what number is the reciprocal??? ? ?

61. A Fraction is a fractional unit, or a collection of fractional units; thus, one-half, two-thirds, four-ninths, etc., are fractions.

Fractions may be written and read as shown below:

Fractions written in this manner are called Common Fractions.

Common fractions are expressed by two numbers, one written above the other, with a line between them. The number below the line is called the Denominator, and the one above it is called the Numerator. Both numerator and denominator are called Terms of the fraction.

The denominator indicates the number of equal parts into which the unit is divided, and the numerator shows how many of these parts are taken. Thus, in the fraction. , the denominator indicates that 1 is divided into 4 equal parts, and the numerator shows that 3 of these are taken.

1. If 1 is divided into 7 equal parts, what is 1 part called? What are 4 parts called? If 1 is divided into 11 equal parts, what are 5 of them called? 6 of them?

2. If 1 yard is divided into 8 equal parts, what is 1 part called? 3 parts? 5 parts? 7 parts? How many eighths of an apple are there in 1 apple? in 3 apples? in 7 apples? How many elevenths in 6? in 11? in 13? in 17? in 19? How many twelfths in 5? in 11? in 13? in 15? in 21 ?

3. In the fraction, what is the denominator? What is the numerator? What is the fractional unit? How many times is this unit taken? How many times are §? Write the following fractions:

1. Seven-eighths.

2. Four-tenths.

3. Nine-twentieths.

4. Eleven-hundredths.

5. Thirteen-twenty fifths
6. Sixty-thousandths.

Read the following fractions:

10 8

125 82

1., 11, 5, 5, H, foo, 100%, #8, 117, H8, 14, 14.

19, 124

62. A fraction is equal to its numerator divided by its denominator. Thus, is equal to 34; for, if each of the units in 3 is divided into four equal parts, we shall have 12 such parts, each equal to, that is, we shall have 12 fourths; but 12 fourths divided by 4 is equal to 3 fourths, or to, that is, 34 is equal to .

63. A fraction may be regarded either as a number, or as an indicated division:

1o. Regarded as a number, the unit is fractional and equal to the reciprocal of the denominator. Thus, is a collection of 3 units, each equal to 1, that is, 3 × = 1. 2o. Regarded as an indicated division, the numerator is the dividend and the denominator is the divisor. Thus, = 3÷4.

MENTAL EXERCISES.

1. In the fraction $, what is the fractional unit? How many times is it taken ? What is 7 times $? 2. Is there any difference between of 1 pound and f of 9 pounds? of 1 dollar, is what part of 4 dollars i How does of 1 dollar differ from of 4 dollars?

DEFINITIONS.

64. A Proper Fraction is one in which the numera tor is less than the denominator; as, 2, .

An Improper Fraction is one in which the numera For is equal to, or greater than the denominator; as, †,§.

NOTE.-If the numerator is less than the denominator, the fraction is less than 1; if the numerator is equal to the denominator, the fraction is equal to 1; if the numerator is greater than the denominator, the fraction is greater than 1.

A Simple Fraction is one, both of whose terms are whole numbers; as,, .

A Mixed Number is a number composed of an integral and of a fractional part; as, 21, 34.

A Complex Fraction is one that has at least one of 子 3 15 7

its terms fractional; as, 5' 14' (1)' 81

NOTE.-A whole number may be regarded as a fraction whose denominator is 1. Thus, 8 = .

FUNDAMENTAL PRINCIPLES.

65. Because the numerator shows how many times the fractional unit is taken, we have the following principles: 1°. Multiplying the numerator of a fraction by any number is equivalent to multiplying the fraction by that number.

2o. Dividing the numerator of a fraction by any number is equivalent to dividing the fraction by that number.

Because the denominator shows the number of equal parts into which we divide the unit 1 to obtain the frac tional unit, we have the following principles:

3°. Multiplying the denominator of a fraction by any